The Probabilities of Pennies

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The Probabilities of Pennies

Ever find a wheat penny, one of those pennies with pieces of wheat on the back instead of the Lincoln Memorial? When I was little, I can remember finding them periodically, and always thinking that this was some sort of special occasion. But how special is it to find a coin of a certain vintage? More generally, many coin collectors start out trying to collect pennies from every year within a given range, just like how many people try to collect quarters from every US state. Can we predict how long this will take to be successful?

Back in 1999, Shiyong Lu, now at Wayne State University, and Steven Skiena, at SUNY Stonybrook, set out to calculate this. They began by recognizing that the probabilities of finding different coins are not equal. The probability of finding a coin from a given year is dependent on its mintage—how many coins of that type were issued that year—how long the coin has been in circulation, and something called collector pressure. The higher the mintage, the higher the probability of finding a coin. However, the higher the age, the lower the chances, because coins get taken out of circulation due to getting lost, whether behind furniture or otherwise. Collector pressure, the final factor, refers to how much collectors try to find a given coin. The higher the collector pressure, the lower the chance of finding that coin, because other collectors are trying to also obtain it and are taking it out of circulation.

For pennies, if you look at coins only from 1959 or newer, the year when wheat pennies were replaced with pennies with the Lincoln Memorial that most of us are familiar with, we can eliminate the difficulty of dealing with collector pressure. So Lu and Skiena set out to calculate the number of pennies needed to collect before they received a complete set of pennies from 1959 to 1997. First, they examines some pennies and calibrated their model to understand how age affects the number of coins in circulation, given their data on mintage.

They then inputted these values into an equation taken from something known as the Coupon Collector's Problem, a well-known problem in probability that can be used to understand our penny question (since collecting coupons is mathematically similar to collecting coins). Running the numbers, Lu and Skiena found that 684 pennies must be collected in order to get one from every year. When they tried this, they got all years from 1959 to 1997 in only 630 coins, not too far from what was expected!

As they concluded with the help of probability, "Filling a penny album remains an affordable goal for children young and old."